Multivariate Representations of Univariate Marked Hawkes Processes

1. Problem Statement

Univariate marked Hawkes processes are widely used to model self-exciting phenomena where events have associated "marks" (e.g., earthquake magnitudes, social media post types, or trade sizes). A significant challenge in modeling these processes is the assumption of mark-separability.

• The Issue: Many standard models (like the ETAS model for earthquakes) assume the ground intensity (time dynamics) and the mark distribution (magnitude/size) are separable, meaning the intensity function factors into a time component and a mark component. However, real-world data often exhibits non-separability, where the history of events influences both the timing and the magnitude of future events in a coupled manner.
• The Difficulty: While non-separable models are theoretically superior, they are computationally intractable for Maximum Likelihood Estimation (MLE) due to slow convergence and high computational costs. Existing alternatives, such as Neural Hawkes processes, offer flexibility but lack parameter interpretability and established statistical properties (consistency, identifiability). Other non-parametric approaches often fail to provide guarantees on approximation quality.

2. Methodology

The authors propose a novel framework that approximates a univariate marked Hawkes process using a multivariate unmarked Hawkes process.

• Core Concept: The method involves partitioning the continuous mark space $\mathcal{M}$ into $K$ disjoint measurable sets (bins) $\{A_1, \dots, A_K\}$.
• Transformation:
  • Events falling into bin $A_i$ are treated as events of "type $i$" in a multivariate system.
  • The original univariate marked process is re-parameterized as a $K$-dimensional multivariate Hawkes process where each component corresponds to a specific mark range.
  • The conditional intensity for component $i$ is defined as:
    $$\lambda_i(t) = \lambda_{0i} + \sum_{j=1}^K \sum_{t_{\ell,j} < t} \alpha_{ij} g_{ij}(t - t_{\ell,j})$$
    Here, $\alpha_{ij}$ captures the cross-excitation between mark bin $j$ and mark bin $i$, effectively modeling the non-separable interaction between mark history and current intensity.
• Approximation Strategy: As the number of partitions $K$ increases, the multivariate representation converges to the true non-separable univariate marked process. This allows the use of standard parametric estimation techniques (MLE) designed for multivariate Hawkes processes on data that was originally univariate and marked.

3. Key Contributions

The paper makes three primary theoretical contributions:

1. Existence of $L_1$ Approximation:
   The authors prove that for any realization of a non-separable univariate marked Hawkes process, there exists a multivariate unmarked representation such that the $L_1$ difference between their conditional intensity functions can be made arbitrarily small by increasing the number of components $K$. This holds under mild assumptions (existence of a positive mark density, continuity of the productivity function, and integrability of the kernel).

2. Stationarity Preservation:
   The paper demonstrates that if the target univariate marked process is stationary, the approximating multivariate representation is also stationary. This is crucial for ensuring the model does not explode (i.e., the expected cluster size remains finite) and allows for valid frequentist forecasting.

3. Parameter Identifiability:
   The authors establish that the parameters of the multivariate representation ($\lambda_{0i}, \alpha_{ij}, \beta_{ij}$) are identifiable. This means that distinct parameter vectors produce distinct probability structures, provided certain regularity conditions (like linear independence of kernel functions and the occurrence of events in all mark bins) are met.

4. Results

• Theoretical Validation:

  • Special Case (Mark-Separable): The authors show that if the underlying process is mark-separable, the MLE of the multivariate representation's background rates corresponds exactly to a histogram estimator of the mark density. This connects the method to well-understood non-parametric estimation theory.
  • General Case (Non-Separable): Theorems 3.2 and 3.3 rigorously prove the convergence and stationarity properties for general non-separable processes.
  • Identifiability: Theorem 3.5 proves that the parameters are identifiable, addressing a common concern when mapping continuous spaces to discrete types.

• Simulation Study:

  • The authors simulated a stationary univariate marked Hawkes process with an exponential kernel and a uniform mark distribution (chosen as a "worst-case" scenario due to maximum entropy).
  • They fitted multivariate representations with varying numbers of components ($K=1$ to $6$).
  • Findings: The median absolute error (MAE) of the parameter estimates decreases as the sample size increases. While the error generally increases with $K$ (due to the larger parameter space), the method successfully recovers the underlying dynamics. The study provides heuristic bounds for the error induced by the increased dimensionality of the multivariate model.

5. Significance and Implications

• Bridging Theory and Practice: This work provides a computationally tractable pathway to model complex, non-separable marked point processes without resorting to "black box" neural networks. It retains the interpretability of parametric models.
• Flexibility: The method does not require prior knowledge of the mark distribution's functional form. It can handle unknown, non-stationary, or non-separable mark distributions by simply increasing the granularity of the partition ($K$).
• Statistical Rigor: Unlike Neural Hawkes processes, this approach offers guarantees on stationarity and parameter identifiability, making it suitable for rigorous statistical inference and forecasting in fields like seismology, finance, and epidemiology.
• Future Directions: The authors suggest that this framework could be extended to kernel density estimators for higher-dimensional mark spaces and that sparsity-inducing regularization could be applied to reduce the parameter space complexity.

In summary, the paper establishes a robust mathematical bridge between univariate marked and multivariate unmarked Hawkes processes, offering a flexible, interpretable, and theoretically sound tool for analyzing complex event data with coupled temporal and magnitude dynamics.